A slow transient diffusion in a drifted stable potential

We consider a diffusion process $X$ in a random potential $\V$ of the form $\V_x = \S_x -\delta x$ where $\delta$ is a positive drift and $\S$ is a strictly stable process of index $\alpha\in (1,2)$ with positive jumps. Then the diffusion is transient and $X_t / \log^\alpha t$ converges in law towards an exponential distribution. This behaviour contrasts with the case where $\V$ is a drifted Brownian motion and provides an example of a transient diffusion in a random potential which is as "slow" as in the recurrent setting.